Methods of characterizing ion-exchanged chemically strengthened glasses containing lithium

ABSTRACT

Methods of characterizing ion-exchanged chemically strengthened Li-containing glasses include: a) measuring a mode spectrum of the glass sample; b) using the mode spectrum, estimating a first contribution to the center tension associated with a spike region and estimating a second contribution to the center tension due to a deep region only, wherein the deep region is assumed to follow a power-law stress profile; and c) determining a total center tension by adding of the first and second contributions to the center tension. The methods can be used for quality control during manufacturing of glass samples by comparing the total center tension to a center tension specification that provides optimum strength and durability.

CROSS-REFERENCE TO RELATED APPLICATIONS

This Application is a continuation of U.S. patent application Ser. No.15/171,490 filed on Jun. 2, 2016, which claims the benefit of priorityunder 35 USC § 119(e) of U.S. Provisional Patent Application No.62/170,919, filed on Jun. 4, 2015, and which is incorporated byreference herein.

FIELD

The present disclosure relates to chemically strengthened glass, and inparticular relates to methods of characterizing ion-exchanged chemicallyglasses containing lithium.

BACKGROUND

Chemically strengthened glasses are glasses that have undergone achemical modification to improve at least one strength-relatedcharacteristic, such as hardness, resistance to fracture, etc.Chemically strengthened glasses have found particular use as coverglasses for display-based electronic devices, especially hand-helddevices such as smart phones and tablets.

In one method, the chemical strengthening is achieved by an ion-exchangeprocess whereby ions in the glass matrix are replaced by externallyintroduced ions, e.g., from a molten bath. The strengthening generallyoccurs when the replacement ions are larger than the native ions (e.g.,Na+ ions replaced by K+ ions). The ion-exchange process gives rise to arefractive index profile that extends from the glass surface into theglass matrix. The refractive index profile has a depth-of-layer or DOLthat defines a size, thickness or “deepness” of the ion-diffusion layeras measured relative to the glass surface. The refractive index profilealso defines a number of stress-related characteristics, includingstress profile, surface stress, center tension, birefringence, etc. Therefractive index profile defines an optical waveguide when the profilemeets certain criteria.

Recently, chemically strengthened glasses with a very large DOL (andmore particularly, a large depth of compression) have been shown to havesuperior resistance to fracture upon face drop on a hard rough surface.Glasses that contain lithium (“Li-containing glasses”) can allow forfast ion exchange (e.g., Li+ exchange with Na+ or K+) to obtain a largeDOL. Substantially power law (e.g., substantially parabolic) stressprofiles are easily obtained in Li-containing glasses, where theion-exchange concentration profile of Na+ connects in the central planeof the substrate, shrinking the traditional central zone of thedepth-invariant center tension to zero or negligible thickness. Theassociated stress profiles have a predictable and large depth ofcompression, e.g., on the order of 20% of the sample thickness, and thisdepth of compression is quite robust with respect to variations in thefabrication conditions.

An example power law stress profile of particular commercial importanceis a near-parabolic (substantially parabolic) profile for the deepregion that joins to a “spike” portion near the surface. This spikeportion (“spike”) is particularly helpful in preventing fracture whenglass is subjected to force on its edge (e.g., a dropped smart phone) orwhen the glass experiences significant bending. The spike can beachieved in Li-containing glasses by ion exchange in a bath containingKNO₃. It is often preferred that the spike be obtained in a bath havinga mixture of KNO₃ and NaNO₃ so that Na+ ions are also exchanged. The Na+ions diffuse faster than K+ ions and thus diffuse at least an order ofmagnitude deeper than the K+ ions. Consequently, the deeper portion(region) of the profile is formed mainly by Na+ ions and the shallowportion of the profile is formed mainly by K+ ions.

In order for chemically strengthened Li-containing glasses to becommercially viable as cover glasses and for other applications, theirquality during manufacturing must be controlled to certainspecifications. This quality control depends in large part on theability to control the ion-exchange process during manufacturing, whichrequires the ability to quickly and non-destructively measure therefractive index (or stress) profiles.

Unfortunately, the quality control for glasses with spike stressprofiles is wanting due to the inability to adequately characterize theprofiles in a non-destructive manner. This inability has mademanufacturing of chemically strengthened Li-containing glasses difficultand has slowed the adoption of chemically strengthened Li-containingglasses in the market.

SUMMARY

An aspect of the disclosure is directed to methods of characterizingchemically strengthened Li-containing glasses having a surface stressspike, such as produced by an ion-exchange process whereby Li+ isexchanged with K+ and Na+ ions (i.e., Li+⇔K+, Na+). The methods resultin a measurement of the surface compression and the depth of the spike,and its contribution to the center tension, as well as the compressionat the bottom of the spike, and the total center tension.

The method is preferably carried out to obtain a commercially importantstress profile. An example of such a profile follows a power law (e.g.,substantially parabolic) in most of the interior (“deep region”) of thesubstrate other than the spike region (“spike”) adjacent the substratesurface. The spike is generally formed by the slower diffusion (and thusshallower) K+ ions while the power-law portion is formed by the faster(and thus deeper) diffusing Na+ ions. The method allows for confirmationthat the profile has reached the power-law, e.g., has a self-consistencycheck. The method can also include performing quality control of theglass samples being process. Such quality control is important for acommercially viable manufacturing process.

An aspect of the method involves measuring the mode lines or fringespectrum of the K+ spike layer of the glass sample using aprism-coupling system to calculate surface stress (CS), depth of thespike (DOS), and the center tension (CT) contribution of the spike. Inaddition, a shift of an intensity transition corresponding to index atthe depth of the bottom of the spike is measured, providing an estimateof the stress at the bottom of the spike. The depth of the spike(DOS_(sp)) and the depth of layer (DOL) is used to verify that the restof the profile is in the power-law regime. Then the total CT isestimated assuming that the profile is follows a power law, taking intoaccount the CT contribution of the spike. If the profile is expected todeviate moderately from a power law, the expected deviation can beaccounted for by an adjustment factor in the target CT. If necessary,the depth of compression (DOC) is also estimated, taking into accountthe reduction of DOC resulting from the spike, relative to the DOC of apower-law profile.

In another aspect of the method, relationship is defined between themeasurement wavelength and the target properties of the Na+-enricheddeeper portion of the profile to help ensure a precise measurement ofthe compression at the bottom of the spike, and the most preciseestimate of the CT and frangibility.

In another aspect of the method, a prism-coupling measurement of themode spectrum is combined with a measurement of the weight gain, ormeasurements of the sample thickness and the sample weight, to estimatethe amount of Na+ diffused in the sample.

Advantages of the methods disclosed herein include that they arenon-destructive and high-throughput, and can provide a measurement ofcritical parameters of the stress profile with high precision. Theseparameters include the compressive stress CS, depth of spike DOS, depthof layer DOL, center tension CT and frangibility status, which is basedon the center tension CT.

Additional features and advantages are set forth in the DetailedDescription that follows, and in part will be readily apparent to thoseskilled in the art from the description or recognized by practicing theembodiments as described in the written description and claims hereof,as well as the appended drawings. It is to be understood that both theforegoing general description and the following Detailed Description aremerely exemplary, and are intended to provide an overview or frameworkto understand the nature and character of the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings are included to provide a furtherunderstanding, and are incorporated in and constitute a part of thisspecification. The drawings illustrate one or more embodiment(s), andtogether with the Detailed Description serve to explain principles andoperation of the various embodiments. As such, the disclosure willbecome more fully understood from the following Detailed Description,taken in conjunction with the accompanying Figures, in which:

FIG. 1A is an elevated view of an example DIOX glass substrate in theform of a planar substrate;

FIG. 1B is a close-up cross-sectional view of the DIOX substrate of FIG.1A as taken in the x-z plane and that illustrates the doubleion-exchange process that takes place across the substrate surface andinto the body of the substrate;

FIG. 1C schematically illustrates the result of the DIOX process thatforms the DIOX substrate;

FIG. 2 is a representation of an example refractive index profile n(z)for the DIOX substrate illustrated in FIG. 1C;

FIG. 3 is a schematic representation of an example measured modespectrum of a Li-containing glass formed by an ion-exchange processusing a mixture of NaNO₃ and KNO₃, with the mode spectrum including TM(top) and TE spectra (bottom), and also showing profile measurementparameters as explained below;

FIG. 4 is a plot of the compressive stress CS (MPa) versus a normalizedposition coordinate z/T, showing the model stress profile (solid line)for a sample chemically strengthened Li-containing glass that hasundergone a K+ and Na+ ion exchange, wherein the dashed line representsthe model profile for Na+ diffusion only, noting that the model profilehas ion exchange taking place at two surfaces that respectively resideat z/T=−0.5 and +0.5;

FIG. 5 is a plot of the stress (MPa) versus a normalized positioncoordinate z/T showing separate plots for the spike portion, the longdiffused power-law portion and the spike plus power-law portion of thestress profile;

FIG. 6 is a schematic representation of a measured mode spectrum showingthe TE and TM mode spectra for an example chemically strengthenedLi-containing glass sample; and

FIGS. 7A and 7B are plots of the intensity I versus distance x′ alongthe mode spectrum for the mode lines or fringes of TE and TM modespectra, respectively, of FIG. 6.

DETAILED DESCRIPTION

Reference is now made in detail to various embodiments of thedisclosure, examples of which are illustrated in the accompanyingdrawings. Whenever possible, the same or like reference numbers andsymbols are used throughout the drawings to refer to the same or likeparts. The drawings are not necessarily to scale, and one skilled in theart will recognize where the drawings have been simplified to illustratethe key aspects of the disclosure.

The claims as set forth below are incorporated into and constitute partof this Detailed Description.

In the discussion below, the term “power law” refers to a function thatin an example follows the general form f(x)=ax+bx²+cx³+dx⁴ . . . , wherea, b, c and d are coefficients.

FIG. 1A is an elevated view an example glass substrate in the form of aplanar ion-exchanged substrate 20 that has a body 21 and a (top) surface22, wherein the body has a base (bulk) refractive index n_(s), a surfacerefractive index no and a thickness T in the z-direction. FIG. 1B is aclose-up cross-sectional view of ion-exchanged substrate 20 as taken inthe y-z plane and illustrates an example double ion-exchange (DIOX)process that takes place across surface 22 and into body 21 in thez-direction.

In the DIOX process discussed in connection the method disclosed herein,two different types of ions Na+ and K+ replace another different ion Li+that is part of the glass body 21. The Na+ and K+ ions can be introducedinto the glass body 21 either sequentially or concurrently using knownion-exchange techniques. As noted above, the Na+ ions diffuse fasterthan the K+ ions and thus go deeper into the glass body 21. This has aneffect on the resulting refractive index profile and stress profile, asdiscussed below.

FIG. 1C is a schematic diagram of the resulting DIOX process, and FIG. 2is a representation of an example refractive index profile n(z) forsubstrate 20 having undergone the DIOX process and having a refractiveindex profile such as illustrated in FIG. 1C. The corresponding stressprofile can be represented by σ(z). The refractive index profile n(z)includes a first “spike” region R1 associated with the shallowerion-exchange (K+ ions) and that has a depth D1 into body 21 that definesa “depth-of-layer for the spike” denoted hereinafter as DOL_(sp). Therefractive index profile n(z) also includes a second region R2associated with the deeper ion-exchange (Na+ ions) and that has a depthD2 that defines the depth-of-layer (DOL).

The deeper second region R2 may be produced in practice prior to theshallower region. The region R1 is adjacent substrate surface 22 and isrelatively steep and shallow, whereas region R2 is less steep andextends relatively deep into the substrate to the aforementioned depthD2. In an example, region R1 has a maximum refractive index no atsubstrate surface 22 and steeply tapers off to an intermediate indexn_(i), while region R2 tapers more gradually from the intermediate indexdown to the substrate (bulk) refractive index n_(s). The portion of therefractive index profile n(z) for region R1 represents a “spike” in therefractive index having a depth DOS.

FIG. 3 is schematic representation of an example measured mode spectrum50 of a Li-containing glass formed by an ion-exchange process using amixture of NaNO₃ and KNO₃, with the mode spectrum including TM and TEspectra 50TM and 50TE (upper and lower portions, respectively) withrespective mode lines 52TM and 52TE. The glass type is 196HLS with afictive temperature of 638° C. The glass was subjected to a Li+⇔K+, Na+ion-exchange process by placing the glass sample in a bath having 60 wt% KNO₃ and 40 wt % NaNO₃ at 390° C. for 3 hours.

As is known in the art, the fringes or mode lines 52TM and 52TE in themode spectrum can be used to calculate surface compression or“compressive stress” CS and depth of layer DOL associated with anion-exchange layer that forms an optical waveguide. In the presentexample, the mode spectrum 50 was obtained using a commerciallyavailable prism-coupling system, namely the FSM6000L surface stressmeter (“FSM system”), available from Luceo Co., Ltd. of Tokyo, Japan.Example prism-coupling systems suitable for use for carrying out themethods disclosed herein are also described in U.S. Patent ApplicationsNo. 2014/0368808 and 2015/0066393, which are incorporated by referenceherein.

The measured values of CS and DOL were 575 MPa and 4.5 microns,respectively. These are the parameters of the K+ enriched layer or spikeregion R1 adjacent sample surface 22. The auxiliary vertical dashedlines on the left hand side of the spectrum were added to FIG. 3 andshow positions in the spectrum in which the conventional FSM systemassigns to correspond to the surface index, one for TM, and one for TE.The difference in these positions is proportional to the surface stressor compressive stress CS, which is denoted “CS_(tot)” in FIG. 3 toindicate the total or maximum amount of surface stress. These positionsare also used in the calculation of DOL.

The present inventors have recognized that in the mode spectrum 50 for achemically strengthened Li-containing glass having undergone a Li+⇔K+,Na+ ion exchange, the position of transition from the bright to the darkportion of the spectrum, observed after the last fringe 52 in thespectrum that corresponds to the highest-order guided mode, is shiftedin the TE spectrum 50TE as compared to the TM spectrum 50TM. This shiftis proportional to the compressive stress CS at the depth at which theK+ concentration in spike region R1 decreases approximately to theconstant-level concentration originally in the substrate (e.g., thespatially constant concentration in the glass matrix that makes upsubstrate body 21). The effective index of the transition corresponds tothe effective index that occurs at the depth of a characteristic “knee”or transition in the stress profile, and the shift of the transitionbetween the TE and TM spectra is proportional to the compressive stressat the depth of the knee (i.e., at the depth of the spike D1=DOL_(sp)).

The measurements of the mode spectrum 50 provided by the K+ penetrationresulting from ion exchange, along with the shift in the intensitytransition (e.g., transition from total internal reflection (TIR) topartial reflection as defined by the critical reflection angle in theprism-coupler system used), can be combined and used for effectivequality control of a family of stress profiles that provide superiorresistance to fracture. The profiles of this family are similar in shapeto a power-law profile (region R2) with a spike (region R1). The spikeis small in thickness when compared to the substrate thickness T. Forexample, the spike may be 10 microns deep (i.e., DOL_(sp)=10 microns),while the substrate may be T=800 microns thick. The spike may have ashape similar to a complementary error function (erfc) shape, but mayalso be similar to a linear depth distribution, Gaussian depthdistribution, or another distribution. The main features of the spikeare that it is a relatively shallow distribution and it providessubstantial increase of surface compression over the level ofcompression at the bottom of the spike as defined by DOL_(sp).

FIG. 4 is a plot of the compressive stress CS (MPa) versus a normalizedposition coordinate z/T, showing the model stress profile (solid) for asample chemically strengthened Li-containing glass that has undergone aK+ and Na+ ion exchange, wherein the dashed line represents the modelprofile for Na+ diffusion only (note that the model profile has ionexchange taking place at two surfaces that respectively reside atz/T=−0.5 and +0.5). The profile has a parabolic deep portion or regionR2 and a surface spike portion or region (or just “spike”) R1.

In the present disclosure, the assumed convention is that compressivestress CS is positive and tensile stress is negative. The model profileof FIG. 4 has a linear spike R1 added on top of a deep quadratic profileR2. Another feature of the spike is also recognized from FIG. 4, namelythat the typical slope of the stress distribution in the spike R1 issignificantly higher than the typical slope in the deep portion R2 ofthe profile, which is assumed to follow a power law, and can beapproximated as parabolic for the purposes of making measurements thatcan be used for quality control.

The profiles that are adequately approximated as parabolic for thepurposes of the quality-control measurement method of the presentdisclosure have a distribution of Na+ ions such that over the range ofdepths from the bottom of the spike to the center of the thickness(i.e., from z=DOL_(sp) to T/2), the minimum Na+ concentration is at thecenter of the thickness (i.e., at x/L=0 in FIG. 4), and the increase inconcentration at the bottom of the spike relative to the minimumconcentration at the center of thickness is approximately 3 to 7 timeshigher than the increase in concentration at a depth equal to onequarter of the thickness, i.e., T/4. In some cases, the increase in Na+concentration anywhere within the depth of the spike relative to theconcentration in the center of the substrate may be 3 to 7 times higherthan the increase in Na+ concentration (relative to the centerconcentration) at depth equal to one quarter of the thickness.

In one embodiment of the method, the compressive stress CS_(knee) andthe depth of the spike are measured using a conventional prism-couplingmeasurement, e.g., using the aforementioned FSM system. Then a CTcontribution of the spike CT_(SP) is calculated using the equation:

${CT}_{sp} = \frac{{CS}_{SP} \times {DOL}_{sp}}{T - {DOL}_{sp}}$

where T is the aforementioned sample thickness. The contribution tocenter tension of the Na+ profile CT_(par), which is assumed parabolic,is calculated as:

CT_(par)≈|0.5σ_(knee)|

where σ_(knee) is the tensile stress at the knee of the profile, e.g.,at the bottom of the spike and is given by:

$\sigma_{knee} = \frac{\left( {n_{crit}^{TE} - n_{crit}^{TM}} \right)}{SOC}$

where n_(crit) ^(TE) and n_(crit) ^(TM) are the effective indices of thecritical-angle intensity transitions as illustrated on FIG. 1. SOC isthe stress-optic coefficient. Note that since tensile stress is theopposite (or negative) of compressive stress, −σ_(knee)=CS_(knee), sothat the magnitude of these two quantities is the same.

As illustrated by the dashed line of FIG. 2, the assumed power-lawprofile may be considered as an auxiliary profile that does not includethe spike and that extends the profile shape all the way to the surfaceof the sample. This auxiliary profile is force-balanced, having its ownCT, and is hence shifted vertically from the power-law portion of themodel spiked power-law profile. The auxiliary profile has a compressiondepth is given by

${DOC}_{par} = {{0.5\; {T\left( {1 - \frac{1}{\sqrt{3}}} \right)}} \approx {0.2113\; T}}$

The total center tension equals the sum of the contributions of thespike and of the power-law portion:

CT_(tot)=CT_(par)+CT_(sp).

The depth of compression DOC of the spiked power-law profile can becalculated by using the expression:

${DOC} = {{0.5\; {T\left( {1 - \sqrt{\frac{{CT}_{tot}}{3\; {CT}_{par}}}} \right)}} = {{0.5\; {T\left( {1 - \sqrt{\frac{1}{3}\left( {1 + \frac{{CT}_{sp}}{{CT}_{par}}} \right)}} \right)}} \approx {{DOC}_{par}\left( {1 - \frac{{CT}_{sp}}{2{{CT}_{par}\left( {\sqrt{3} - 1} \right)}}} \right)} \approx {{DOC}_{par} - {\frac{0.5\; T}{\sqrt{3}}\frac{{CT}_{sp}}{2\; {CT}_{par}}}}}}$

The approximate expressions at the end of the above equation are validwhen the CT contribution of the spike is significantly smaller than theCT contribution of the auxiliary power-law profile.

In summary, the above-described method includes: 1) measuring the modespectrum due to the spike and estimating a contribution of the spike tothe center tension; 2) using the mode spectrum, measuring the shift ofthe critical angle; 3) estimating a compression at the knee, which isproportional to the contributions of the substantially power-law profileand of the spike.

In addition, a compression depth DOC of the profile may be estimated byusing an exact formula for the model profile, or an approximate formulathat gives the DOC as the DOC of an auxiliary power-law profile (e.g.,parabolic or nearly parabolic) less a small DOC reduction due to thespike.

Furthermore, in one example embodiment, the depth of spike DOL_(sp) isused to verify that the deep portion of the profile (region R2) is in aregime well represented by the power-law profile shape. In particular,as the DOL_(sp) of the spike increases, the penetration of Na+ increasesapproximately in proportion to the DOL_(sp) of the spike. Hence, for aglass where simultaneous in-diffusion of K+ and Na+ is used, a minimumDOL_(sp) can be set for any particular glass thickness, above which thedeep portion of the profile can be considered to follow a power law. Inanother example, an upper limit of the DOL_(sp) may also be imposed toexclude physical profiles that start to deviate substantially from theparabolic model as a result of stress relaxation, which may becomeprominent at very long ion exchange times at high temperature.

The above-described method is a somewhat simplified version of a moreprecise method. The simplification incurs only a minor error when the CTcontribution of the spike is much smaller than the CT contribution ofthe auxiliary power-law profile. The CT contribution of the spike shiftsthe power-law portion vertically by the amount CT_(sp) relative to theauxiliary power-law profile. As a result, the compression at the knee ofthe model spiked profile is actually smaller than the compression of theauxiliary profile at the knee depth by the amount CT_(sp).

Furthermore, there is a minor change in compression of the auxiliarypower-law profile between the surface and the depth of the knee, and,for a force-balanced power-law profile the CT is actually equal to halfof the surface compression.

The following steps represent an example of a more precise method fordetermining the parameters of the model spiked power-law profile fromthe prism-coupling measurements:

-   a) Calculate preliminary

${CT}_{sp}^{(0)} = \frac{\left( {{CS}_{tot} - {CS}_{knee}} \right) \times {DOL}_{sp}}{T - {DOL}_{sp}}$

-   b) Calculate preliminary

${CS}_{par}^{(0)} = \frac{{CS}_{knee} + {CT}_{sp}^{(0)}}{1 - {6\frac{{DOL}_{sp}}{T}} + {6\frac{{DOL}_{sp}^{2}}{T^{2}}}}$

-   c) (Optional alternative to steps 4, 5, and 6) Calculate preliminary

${CT}_{par}^{(0)} = {{0.5\; {CS}_{par}^{(0)}} = {0.5\frac{{CS}_{knee} + {CT}_{sp}^{(0)}}{1 - {6\frac{{DOL}_{sp}}{T}} + {6\frac{{DOL}_{sp}^{2}}{T^{2}}}}}}$and CT_(tot)⁽⁰⁾ = CT_(par)⁽⁰⁾ + CT_(sp)⁽⁰⁾

-   d) Calculate more precise

${CT}_{sp}^{(1)} = \frac{\left( {{CS}_{tot} - {CS}_{par}^{(0)}} \right) \times {DOL}_{sp}}{T - {DOL}_{sp}}$

-   e) Calculate more precise

${{CS}_{par}^{(1)} = \frac{{CS}_{knee} + {CT}_{sp}^{(1)}}{1 - {6\frac{{DOL}_{sp}}{T}} + {6\frac{{DOL}_{sp}^{2}}{T^{2}}}}},{and}$${CT}_{par}^{(1)} = {{0.5{CS}_{par}^{(1)}} = {0.5\frac{{CS}_{knee} + {CT}_{sp}^{(1)}}{1 - {6\frac{{DOL}_{sp}}{T}} + {6\frac{{DOL}_{sp}^{2}}{T}}}}}$

-   f) Calculate more precise CT_(tot) ⁽¹⁾=CT_(par) ⁽¹⁾+CT_(sp) ⁽¹⁾-   g) (Optional)—Can continue iteration, finding more and more precise    values for CT_(sp) and CS_(par) until desired level of convergence    or precision. More than one iteration would rarely be of use, since    actual profiles are unlikely to be precisely power law and have a    precisely linear spike. More than one iteration may be useful in    relatively thin substrates in which the depth of the spike may    represent more than about 3% of the substrate thickness.-   h) (Optional) Determine depth of compression DOC of the profile, for    example using one of the forms of the equation:

${DOC} = {{0.5\; {T\left( {1 - \sqrt{\frac{{CT}_{tot}}{3\; {CT}_{par}}}} \right)}} = {{0.5\; {T\left( {1 - \sqrt{\frac{1}{3}\left( {1 + \frac{{CT}_{sp}}{{CT}_{par}}} \right)}} \right)}} \approx {{DOC}_{par}\left( {1 - \frac{{CT}_{sp}}{2{{CT}_{par}\left( {\sqrt{3} - 1} \right)}}} \right)} \approx {{DOC}_{par} - {\frac{0.5\; T}{\sqrt{3}}\frac{{CT}_{sp}}{2{CT}_{par}}}}}}$

In the case where one is working directly with the DOL as measured byconventional means and not the real depth of layer or position of thespike, a range of approximations may be needed, some of which areempirical in nature.

In many cases, the power-law profile can be estimated as parabolic orsubstantially parabolic, i.e., to the power 2. In other cases,substantially nonlinear diffusion may cause the shape of the deepportion of the stress profile to be best represented by a relativelyhigh power-law profile, e.g., to the power 3, i.e., cubic, or evenhigher. The methods that employ profiles greater than parabolic can canbe used to improve the accuracy with which the quality-control modelrepresents the actual stress profile, and in this way to increase therange of ion exchange conditions over which the quality control methodworks effectively for the particular glass of interest.

For example, for a parabolic profile the ratio of CS to CT by absolutevalue is 2. Similarly, for a cubic profile the corresponding ratio is 3,and for a quartic profile it is 4. A force-balanced profilerepresentation using the power of 3/2 has a CS to CT ratio of 1.8. Notethat for power-function profile where the power is not an even number,the profile is constructed as the absolute value of the power function.In the mentioned examples, the depth of compression for the non-spikedauxiliary profile is about 0.23T for the profile with power 3/2, 0.21Tfor the power-law profile, 0.185T for the cubic profile, and 0.165T forthe quartic profile.

Once a model-profile shape is found to represent the relatively stableshape of the center-connected concentration and stress profile, thecorresponding ratio of CS to CT and DOC estimate can be used for theauxiliary profile in the application of the method of the presentinvention for quality control.

Thus, in a more generic embodiment of the invention, the auxiliary deepportion of the spiked profile may be represented with a symmetricabsolute value of a power function of the distance from the center, andan associated ratio of CS to CT of a force-balanced profile having theshape of the model function is used to estimate the CT from the measuredcompression at the knee or estimated compression at the surface. If thedepth of compression is also sought, the depth of compression of themodel profile may be used as an approximate reference, or the depth ofcompression of the full force-balanced model profile may be calculatedas was done for the example spiked power-law profile. In an example, theratio CS to CT ranges from about 1.8 to about 4, while the DOC for theauxiliary model profile ranges from about 0.16T to about 0.23T.

FIG. 5 is a plot of the stress (MPa) versus a normalized positioncoordinate x/L and shows separate plots for the spike portion, the longdiffused power-law portion and the spike plus power-law portion of thestress profile. Another embodiment uses the parabolic auxiliary profileas illustrated in FIG. 4 and the following representation for theforce-balanced power-law profile (the solid line):

$\begin{matrix}{{\sigma_{1}\left( \frac{z}{L} \right)} = {{\sigma_{1}(0)}\left\lbrack {{- \frac{1}{2}} + {6\left( {\frac{z}{L} - 0.5} \right)^{2}}} \right\rbrack}} & (1)\end{matrix}$

A conventional measurement (e.g., using the aforementioned FSM system)is used to measure “FSM_DOL” of the spike as approximately the diffusiondepth given by 2√{square root over (D,τ)} where D is the diffusioncoefficient and τ is the time of diffusion. Then, using the fact abovewhere ˜1.25×FSM_DOL leads to most of the area of the spike to beincluded, one can get an approximate equation for the CT due to thespike contribution, namely

$\begin{matrix}{{CT}_{spike} = \frac{\left( {{\sigma_{2}^{\prime}(0)} - {\sigma_{2}^{\prime}\left( {1.25{{FSM}\_ {DOL}}} \right)}} \right) \times {1.25{{FSM}\_ {DOL}}}}{T - {{1.25{FSM}}\mspace{11mu} {DOL}}}} & (2)\end{matrix}$

The point σ₂′ (1.25×FSM_DOL) although not equal to is acceptably closeto the CS between the transition between guided modes and continuum inthe spiked lithium glass samples. This point is called the CS_(knee) andis shown in FIG. 3 as a reasonable approximation. It is also reasonableto approximate the stress and the offset due to the contribution of theCT_(spike) deeper inside the glass.

Since the power-law profile will be slowly varying as compared to thespike portion of the profile, it can be assumed that the stress at˜(2-3)×FSM_DOL in the power-law region would not feel the presence ofthe spike. It therefore follows that:

$\begin{matrix}{{{\sigma_{1}\left( \frac{z}{L} \right)} \cong {{\sigma_{1}^{\prime}\left( \frac{z}{L} \right)} + {CT}_{spike}}}{{and},}} & (3) \\{{\sigma_{1}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)} \cong {{\sigma_{1}^{\prime}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)} + {CT}_{spike}}} & (4)\end{matrix}$

Using the parabolic power-law equation in (1), one establish thefollowing relationships:

$\begin{matrix}{{\sigma_{1}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)} = {{{\sigma_{1}(0)}\left\lbrack {{- \frac{1}{2}} + {6\left( {\frac{3{{FSM}\_ {DOL}}}{L} - 0.5} \right)^{2}}} \right\rbrack} \cong {{\sigma_{1}^{\prime}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)} + {{CT}_{spike}\mspace{20mu} {and}}}}} & (5) \\{\mspace{79mu} {{\sigma_{1}(0)} \cong \frac{{\sigma_{1}^{\prime}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)} + {CT}_{spike}}{\left\lbrack {{- \frac{1}{2}} + {6\left( {\frac{3{{FSM}\_ {DOL}}}{L} - 0.5} \right)^{2}}} \right\rbrack}}} & (6)\end{matrix}$

Numerically it can be demonstrated that if one uses a factor 2 insteadof 3, the results are almost the same, in some cases varying by just1%-2% of σ₁(0). Therefore, if one can find the approximated value of

${\sigma_{1}^{\prime}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)}\mspace{14mu} {or}\mspace{14mu} {\sigma_{1}^{\prime}\left( \frac{2{{FSM}\_ {DOL}}}{L} \right)}$

in the FSM, one can use equation (6) to compute the original stress ofthe first stress parabola within this range of error.

In practice one can measure approximately ˜

${\sigma_{1}^{\prime}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)}\mspace{11mu}$

to ˜

$\sigma_{1}^{\prime}\left( \frac{2{{FSM}\_ {DOL}}}{L} \right)$

by examining the stress generated at the transition between guided modesand continuum in the spike on Li-glass samples. This point whereapproximately ˜

${\sigma_{1}^{\prime}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)}\mspace{11mu}$

to ˜

$\sigma_{1}^{\prime}\left( \frac{2{{FSM}\_ {DOL}}}{L} \right)$

can be used as the point CS_(knee) as shown in FIG. 3 as a reasonableapproximation that can be measured by computing the distance betweenthese lines (fringes) in TM and TE spectra and their refractive index.In light of the stress optical coefficient (SOC) of the material thedivision of the index difference at this point per the SOC would lead tothe CS_(knee) stress values.

This in addition to the FSM_DOL and the CS˜σ₂′(0) given by the FSM forthe spike. Therefore CT_(parabola)=CT_(par)˜σ₁(0)/2, and CT_(spike) isgiven in (2) as (repeated for convenience)

$\begin{matrix}{{CT}_{spike} = \frac{\left( {{\sigma_{2}^{\prime}(0)} - {\sigma_{2}^{\prime}\left( {1.25{{FSM}\_ {DOL}}} \right)}} \right) \times {1.25{{FSM}\_ {DOL}}}}{T - {{1.25{FSM}}\mspace{11mu} {DOL}}}} & (2)\end{matrix}$

From there, one can (repeating the previous equations) then compute thetotal center tension equals the sum of the contributions of the spikeand of the power-law portion:

CT_(tot)=CT_(par)+CT_(spike)  (7)

If desired the depth of compression of the spiked power-law profile canbe calculated/estimated by using the expression:

$\begin{matrix}{{DOCtotal} = {{0.5\; {T\left( {1 - \sqrt{\frac{{CT}_{tot}}{3\; {CT}_{par}}}} \right)}} = {{0.5\; T\left( {1 - \sqrt{\frac{1}{3}\left( {1 + \frac{{CT}_{sp}}{{CT}_{par}}} \right)}} \right)} \approx {{DOC}_{par}\left( {1 - \frac{{CT}_{sp}}{2{{CT}_{par}\left( {\sqrt{3} - 1} \right)}}} \right)} \approx {{DOC}_{par} - {\frac{0.5\; T}{\sqrt{3}}\frac{{CT}_{sp}}{2{CT}_{par}}}}}}} & (2)\end{matrix}$

These equations assume that the deep part of the profile isapproximately parabolic in nature and has an added spike near thesurface. Its validity is better matched when the spike is small instress amplitude and not so deep in comparison to the deeper power-lawportion of the profile.

An aspect of the disclosure is the use the algorithm described by theabove equations to process measured mode spectra. FIG. 6 is a schematicrepresentation of a measured mode spectrum showing the TE and TM modespectra based on actual measured mode spectra for an example chemicallystrengthened Li-containing glass sample. In this case using thealgorithm above considering that both σ₂′(1.25×FSM_DOL)˜CS_(knee) andthat

${\sigma_{1}^{\prime}\left( \frac{3{{FSM}\_ {DOL}}}{L} \right)}\;$

˜CS_(knee) as reasonable approximations, one can find that:

FSM_DOL=DOL_(sp)=7.77 um

σ₂′(0)=CSspike=482.6 MPa

CS_(knee)=138.8 MPa

CT_(spike)=4.23 MPa

CS_(parabola)=σ₁′(0)=172.3 MPa

CT_(parabola)=86.14 MPa

CT_(total)=90.4 MPa

DOC_(parabola)=169 um

DOC_(total)=163.4 um

The method of measurement of the compression at the knee as describedabove has some accuracy limitations. For any particular measurementwavelength, there are certain quasi-periodic ranges of the spike DOLwithin which the accuracy of the measurement of the critical-angle shiftis reduced. The cause of this reduction is the appearance of a leakymode with an effective index close to that of the critical angle as theDOL is increased from a range where the critical-angle transition isclear-cut.

In the example of FIG. 3, the signature of such a leaky mode can beobserved in the TM mode spectrum as the extra-dark relatively narrowband immediately on the right-hand-side of the transition from bright todark. When the leaky mode has an effective index very close to thecritical-angle index, the shape of the critical-angle transition ischanged by the leaky mode, resulting in error in the measurement of theshift. When a leaky mode affects only the critical-angle transition forthe TM spectrum, the compression at the knee is slightly over-estimated.When it affects only the critical-angle transition for the TE spectrum,the compression at the knee is slightly under-estimated. When a leakymode affects both the TM and the TE transition, the net error of thecompression at the knee could be positive or negative, and smaller ascompared to the error when the transition for only one polarization issubstantially affected.

Consequently, aspects of the measurement methods disclosed herein aredirected to eliminate or mitigate the error in the measurement of theknee compression that is caused by the presence of a leaky mode. In oneembodiment, the target DOL_(sp) is selected such that the effectiveindex of a leaky mode is not close to the critical-angle transition,neither for the TM nor for the TE polarization. If such a choice ofDOL_(sp) is not available within the range of preferable overall productstress-profile designs, then the measurement wavelength is chosen suchthat a leaky mode at the measurement wavelength is not close to thecritical angle at either polarization. The way the DOL is measuredtraditionally is by assigning a fraction of a mode to the index rangebetween the highest-order guided mode and the critical angle for thepurposes of mode counting, to provide a continuous measured DOL as afunction of increasing diffusion depth by using a continuously varyingnumber of modes.

The equation for the DOL is:

${DOL} = \frac{\left( {{2\; N} - 0.5} \right)\lambda}{3.77\sqrt{n\; \Delta \; n}}$

where n is the approximate average index of the measured waveguidinglayer, and Δn is the difference between the maximum and minimum index ofthe waveguide assumed to have a truncated linear profile. In the case ofthe measurement of the spike, Δn refers to the difference between thesurface index and the index at the knee of the stress profile. N is thecontinuous number of modes. In the example of FIG. 3, N is approximately2.6 modes for the TM spectrum and 2.4 modes for the TE spectrum.

The condition for selecting a preferred target DOL or a preferredmeasurement wavelength is that the non-integer part of the number ofmodes N be between about 0.1 and 0.6 at both the TM and TE polarization.Hence, in one embodiment of the invention, the measurement wavelengthand the depth of the spike are such that the non-integer part of thenumber of modes is between about 0.1 and 0.6 at both polarizations,preferably between 0.2 and 0.55.

In another less restrictive embodiment, a correction for the effect ofthe leaky mode is made during the interpretation of the measured shiftof the critical transition. The non-integer part of each of the TM andTE number of modes is taken into account, and a correction is made thataccounts effect of the leaky mode. To this end, a calibration of theeffect of the leaky mode of each polarization on the measured kneecompression may be used. In an example, a sequence of samples havingDOL_(sp) increasing in small steps is fabricated in the same saltmixture and at the same temperature, and used for the calibration. Theactual compression at the knee decreases monotonically and smoothly withincreasing DOL_(sp), while the estimated compression fluctuates aboveand below the smooth dependence of the actual compression on DOL_(sp) asa result of the error induced by the leaky mode. By tracking thedependence of the fluctuation on the non-integer part of the TM and TEmode numbers, the calibration of the necessary correction is obtained.

In another embodiment, the intensity profile in the vicinity of thecritical angle is analyzed, the contribution of the leaky mode isestimated based on the shape of the intensity profile and thenon-integer mode number, and the estimated contribution is corrected forin the intensity profile to calculate a more accurate position of thecritical angle. This is done independently for the TM and the TEpolarization, and the stress at the knee is estimated after the TM andTE critical angles are found following the application of the intensitycorrections.

In another embodiment, the weight gain of a sample as a result of ionexchange is used in combination with the prism-coupling measurement. Theweight gain may be used to verify that enough Na ions have exchanged forLi ions such that the use of the power-law profile model is valid forquality control. For the purpose, a target acceptable weight gain rangeis prescribed for the ion exchange based on the total surface area ofthe sample and the sample thickness. The weight of representativesamples is measured before and after ion exchange, and thequality-control prism-coupling measurements are considered valid if themeasured weight gain per sample falls in the target range.

In another embodiment of the method, advantage is taken of the precisecontrol of the sample shape, and of individual-sample thicknessmeasurements that are common in some production processes. In this caseit is possible to verify that the sample has had adequate weight gain bysimply measuring the sample thickness with high precision (such as +/−2microns), and measuring the post-ion exchange weight of the sample. Fromthe known shape specification, the measured thickness, and the knowndensity of the pre-ion-exchanged glass, the weight of thepre-ion-exchanged sample is calculated. A correction factor may beapplied that accounts for a typical volume change as a result of ionexchange. The weight gain then is estimated by subtracting from themeasured post-ion-exchange weight the estimated pre-exchanged weight. Ifthe weight gain falls within the target range, the profile is deemedadequately represented by the quality-control model profile, and theprism-coupling QC measurement is considered valid.

It will be apparent to those skilled in the art that variousmodifications to the preferred embodiments of the disclosure asdescribed herein can be made without departing from the spirit or scopeof the disclosure as defined in the appended claims. Thus, thedisclosure covers the modifications and variations provided they comewithin the scope of the appended claims and the equivalents thereto.

1-13. (canceled)
 14. A method of improving functionality of aprism-coupling system to measure a center tension CT in a chemicallystrengthened Li-containing glass sample having a surface and a body witha thickness T, wherein the glass sample has been strengthened byperforming an ion-exchange process that exchanges Li+ ions in the glasssample with Na+ and K+ ions, wherein the K+ ions and Na+ ions define astress profile that includes a spike region adjacent the sample surfaceand defined substantially by the K+ ions and a deep region that extendsfrom a bottom of the spike region into the glass body and definedsubstantially by the Na+ ions, the spike region and the deep region arejoined at a knee having a knee depth DOL_(sp), the method comprising: a)measuring a mode spectrum of the glass sample using the prism-couplingsystem; b) estimating from the mode spectrum: i) a surface compressivestress CS_(sp) produced by the spike region; ii) a spike center tensionCT_(sp) for the spike region only using the relationshipCT_(sp)=[CS_(sp)×DOL_(sp)]/[T−DOL_(sp)]; iii) a knee stress CS_(knee) atthe knee, and iv) a deep region center tension CT_(par) as produced bythe deep region only using the knee stress CS_(knee) and a parabolicprofile approximation for the deep region center tension CT_(par); andc) estimating the center tension CT of the glass sample by therelationship CT_(sp)+CT_(par).
 15. The method according to claim 14,further comprising performing quality control of the formation ofchemically strengthened Li-containing glass samples by performing actsa) through c) to non-destructively measure the center tension ofmultiple glass samples and adjusting the ion-exchange process so thatthe center tension of subsequent glass samples falls within a centertension specification.
 16. The method according to claim 14, whereinacts b) and c) are carried out in a processor of the prism-couplingsystem using instructions embodied in a non-transitory computer-readablemedium that controls the processor.
 17. The method according to claim14, wherein the deep region center tension CT_(par)≈|0.5CS_(knee)|. 18.A method of improving functionality of a prism-coupling system tomeasure a center tension CT in a chemically strengthened Li-containingglass sample having a surface and a body with a thickness T, wherein theglass sample has been strengthened by performing an ion-exchangeprocess, the glass sample has a stress profile that includes a spikeregion adjacent the sample surface and a deep region that extends from abottom of the spike region into the glass body, the method comprising:a) measuring a center tension CT in the glass sample; b) measuring amode spectrum of the glass sample using the prism-coupling system, theprism-coupling system having a controller; c) using the controller,estimating from the mode spectrum, a first center tension for the spikeregion only and estimating a second center tension for the deep regiononly, including assuming a power-law stress profile for the deep region;and d) estimating in the controller a total center tension CT for theglass sample by adding the first and second center tensions.
 19. Themethod of claim 18, further comprising: e) comparing in the controllerthe determined center tension CT of act c) to a center tensionspecification; and f) adjusting the ion-exchange process based on theact e) of comparing of the determined center tension CT to the centertension specification for the chemically strengthened first glasssample.
 20. The method of claim 18, wherein the power-law stress profileis substantially parabolic.
 21. The method of claim 18, wherein stressprofile is formed using an ion-exchange of Li+ ions in the glass with K+and Na+ ions.
 22. The method of claim 18, wherein measuring the modespectrum includes capturing the mode spectrum on a photodetector of theprism-coupling system.
 23. The method of claim 18, further includingmeasuring at least one additional stress-related characteristic of theglass sample.
 24. The method of claim 18, wherein the substrate has athickness T, the spike region has a surface compressive stress CS, andthe spike region and the deep region are joined at a knee having a kneedepth DOL_(sp), and wherein: the first contribution to the centertension is defined by a spike center tension defined byCT_(sp)=[CS_(sp)×DOL_(sp)]/[T−DOL_(sp)].
 25. The method of claim 24,wherein the knee has a knee stress CS_(knee) and the second contributionto the center tension is a deep region center tension CT_(par) definedby CT_(par)≈|0.5CS_(knee)|.
 26. A method of improving the functionalityof a prism-coupling system having a controller to perform qualitycontrol of chemically strengthened Li-containing glass samples, themethod comprising: a) measuring a mode spectrum of the glass sampleusing the prism-coupling system; b) using the controller, determiningthe non-integer part of the number of modes of the mode spectrum; and c)comparing the non-integer part of the number of modes to a non-integerpart of the number of modes specification to determine whether the glasssample satisfies a quality control specification, wherein thenon-integer part of the number of modes specification is from greaterthan or equal to 0.1 to less than or equal to 0.6.
 27. The method ofclaim 26, wherein the non-integer part of the number of modesspecification is from greater than or equal to 0.2 to less than or equalto 0.55.
 28. The method of claim 26, wherein the non-integer part of thenumber of modes is determined for TM polarization and the TEpolarization.
 29. The method of claim 28, wherein the non-integer partof the number of modes for both the TM polarization and the TEpolarization are compared to the non-integer part of the number of modesspecification.
 30. A method of improving functionality of aprism-coupling system to measure a depth of compression DOC in achemically strengthened Li-containing glass sample having a surface anda body with a thickness T, wherein the glass sample has beenstrengthened by performing an ion-exchange process that exchanges Li+ions in the glass sample with Na+ and K+ ions, the method comprising: a)measuring a mode spectrum of the glass sample using the prism-couplingsystem; b) estimating the DOC from the mode spectrum.
 31. The method ofclaim 30, wherein estimating the DOC from the mode spectrum comprisesestimating: i) a spike center tension CT_(sp) for the spike region only,and ii) a deep region center tension CT_(par) as produced by the deepregion only.